Lemma 87.30.3. Let $A \to B$ be a local map of local Noetherian rings such that

$A \to B$ is flat,

$\mathfrak m_ B = \mathfrak m_ A B$, and

$\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$

Then the base change functor from the category (87.30.0.1) for $(A, \mathfrak m_ A)$ to the category (87.30.0.1) for $(B, \mathfrak m_ B)$ is an equivalence.

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